document.write( "Question 413900: write as a single logarithium
\n" ); document.write( "1/2logb16+(2logbx - log bxy) \n" ); document.write( "
Algebra.Com's Answer #290564 by jsmallt9(3759)\"\" \"About 
You can put this solution on YOUR website!
It is only because of the \"b16\" and that I am able to guess that \"b\" is base and 16 the argument. (If the base was 10 and the argument was b times 16 then the argument would normally be written as \"16b\" not \"b16\".) In the future pleaseUse more explicit language like \"1/2 the base b log of 16...\"; or
  • Click on \"Show source\" so you can see and learn how I am able to show logarithms with bases properly.

    • \n" ); document.write( "\"%281%2F2%29log%28b%2C+%2816%29%29+%2B+%282log%28b%2C+%28x%29%29+-+log%28b%2C+%28xy%29%29%29\"
    \n" ); document.write( "Like logarithmic terms have logarithms with the same bases and same arguments. Your logarithms all have the same bases, b. But their arguments are different so they are not like terms. And since they are not like terms we cannot just add and/or subtract to combine them.

    \n" ); document.write( "But there is another way to combine logarithmic terms. There are two properties of logarithms which provide another way to combine logarithmic terms:
    • \"log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29\"
    • \"log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29\"
    These properties require logarithms of the same base and coefficients of 1. Only one of your logarithms has a coefficient of 1.

    \n" ); document.write( "Fortunately there is a third property, \"q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29\", which allows us to move a coefficient into the argument as its exponent. So we can use this property to get the coefficients of 1 on the first two logs:
    \n" ); document.write( "
    \n" ); document.write( "Since an exponent of 1/2 means square root, I'm going to rewrite that argument as a square root:
    \n" ); document.write( "
    \n" ); document.write( "And since the square root of 16 is 4:
    \n" ); document.write( "\"log%28b%2C+%284%29%29+%2B+%28log%28b%2C+%28x%5E2%29%29+-+log%28b%2C+%28xy%29%29%29\"
    \n" ); document.write( "Now that the bases are equal and the coefficients are 1's we can start using the first two properties to combine them. We'll start inside the parentheses. We will use the second property which shows us how to combine two logs with a \"-\" between them:
    \n" ); document.write( "\"log%28b%2C+%284%29%29+%2B+%28log%28b%2C+%28x%5E2%2F%28xy%29%29%29%29\"
    \n" ); document.write( "An \"x\" will cancel in that fraction giving us:
    \n" ); document.write( "\"log%28b%2C+%284%29%29+%2B+%28log%28b%2C+%28x%2Fy%29%29%29\"
    \n" ); document.write( "Now we can use the first property which shows us how to combine logarithms with a \"+' between them:
    \n" ); document.write( "\"log%28b%2C+%284%2A%28x%2Fy%29%29%29\"
    \n" ); document.write( "which simplifies to:
    \n" ); document.write( "\"log%28b%2C+%28%284x%29%2Fy%29%29\"     \n" ); document.write( "
    \n" );